THE BRANCH POINT STRUCTURE OF EXTENSIONS OF INTERIOR BOUNDARIES
BY MORRIS L. MARX
1. Introduction. Let D e E2 be an open set such that D is compact and Bound- ary D is an oriented Jordan curve. A mapping /: D -»■E2 is said to be properly interior iff is light, open, and sense-preserving on D, and a local homeomorphism (relative to D) at each point of Boundary D. A theorem of Stoilow [6, Theorem 4.5, p. 98] states that if/is properly interior, then at each pointy of D there exists a closed two-cell neighborhood A of p on which / is topologically equivalent to w=zn on z SI for some positive integer «. The integer «— 1 is called the multiplicity off at p. A branch point is a point at which the multiplicity is strictly larger than zero. Suppose S : Boundary D -»• E2 is continuous ; then S is an interior boundary if there exists a properly interior/: D -*■E2 such that/[Boundary 7>= 8. The mapping / is said to be an extension of 8 to D. If 8 is an interior boundary and / is any properly interior extension of 8, then / has only finitely many branch points, say zx,..., zr with multiplicities p(l),..., p(r). It is well known that 1 + p(l ) + ■■ ■ + p(r) is always equal to the tangential winding number of 8 [2, Theorem 20.1, p. 71]. What has not been considered is what minimal value r can take for a given interior boundary and what multiplicities can occur for the minimal r. This is part of the problem of determining more about the branch points, poles, and zeroes of light open mappings of the two-cell into the two sphere (see [2, p. 76]). An algorithm is presented here which answers this question for the case that S is a representation of a normal curve. A mapping 8 of an oriented one-dimensional manifold 7 into the complex plane E2 is called a representation ; if 8 possesses a continuous non- vanishing tangent 8' then 8 is called a regular representation. An image point 80 of a regular representation Sisa vertex if there exist exactly two distinct points x and y such that S(x) = 8(y) = S0and if the tangents S'(x) and S'(y) are linearly independent. A regular representation is normal if it has a finite number of vertices and every other image point has but one pre-image. Two representations (regular repre- sentations) 8 and e are equivalent if there exists a sense-preserving homeomorphism c4:7-^7 such that e = So<£ (and 'is continuous and nonvanishing). A regular (normal) curve is then defined to be an oriented curve with a regular (normal) representation. In [4] Titus has given an algorithm to determine if a normal representation S is an interior boundary. His method is to "cut" 8 into two normal representations 8* and 8** in such a way that 8 is an interior boundary if and only if 8* and 8** are
Presented to the Society, August 31, 1967; received by the editors December 16, 1966. 79
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 80 M. L. MARX [April and S* and 8** have less vertices than S. The algorithm presented in this paper is closely related to Titus' ; in fact, his cut of Type IF is unchanged here. The Titus cut of Type I destroys information about branch point structure and so another type of cut is introduced. This new cut algorithm not only answers questions about the branch point structure but can also be used to determine if a normal repre- sentation is an interior boundary. The price paid for this dual capability is increased complexity of the algorithm. 2. Preliminaries. In what follows S will be a representation of a closed curve. Let [S] denote the point set consisting of image points of 8. Let w(8, p) be the index of 8 about a point p not in [8]. The outer boundary of S will be the subset of [8] which is contained in the closure of the unbounded component of the complement of [8]; an outer point p is a point on the outer boundary such that p has but one pre-image. For normal 8 and any nonvertex/» one can define w +(8, p) and w~(8, p) as the larger and smaller winding numbers of points p' near p but not on [8]. An outer point p is positive [negative] ifw +(o,p)=l[w-(8,p)=-l]. Let S be normal and let 8(0) be an outer point where 8 is given by the complex- valued function 8(t) = a(t) + ib(t) with t the usual angle parameter, 0^r<27r. Index the n vertices in the natural way by traversing the curve with increasing t and using consecutively the integers l,2,...,n; thus 8X, 82,..., 8n. (See Figure 1.) The 2« pre-images of the vertices will be denoted by the lower case Roman equivalent of the Greek letter denoting the curve and they will be indexed so that 0 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1968] EXTENSIONS OF INTERIOR BOUNDARIES 81 Figure 1 have isomorphic intersection sequences, they are equivalent ; moreover, if one has an extension to the disk with certain topological properties, so will the other. Hence we consider two representations with isomorphic intersection sequences as interchangeable. Theorem [4, p. 49]. Let 8 be a normal representation defined on a Jordan curve D; let e be normal and defined on the Jordan curve E. Suppose that each curve has a positive outer point and the curves have isomorphic intersection sequences. Then there exists a sense-preserving homeomorphism « of E2 onto E2 taking D onto E such that eh = S (juxtaposition denotes function composition). Suppose 8 and e are representations and x=8(p) and y = &(q) are two points v/ith p License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 82 M. L. MARX [April E2—J. The set Ins J will be called a Jordan domain. If a and b are two points of J, then [a, b] denotes the set of points encountered as / is traversed from a to b in the positive orientation; also (a, b) denotes [a, b]—{a, b}. We will assume from this point on that a properly interior mapping f on D does not map branch points onto f(D —D). There is no loss of generality in this assumption for if / does not have this property, there exists a homeomorphism h: Z>-> D such that /composed with h has the property. For our purposes/and f o hare interchangeable. 3. Representations with A= 1. Let S be a normal representation with a properly interior extension /. Suppose / has r distinct branch points with multiplicities p(l), p(2),..., p(r). The sum 1 +p(l)+ ■■ ■ +p(r) is always equal to the tangential winding number of 8 [2, Theorem 20.1, p. 71]. There remains the question of what minimum value r can take, and in that case, what can the multiplicities be. The results of this section answer that question for the class of representations 8 such that A(8^)= 1 for every vertex S; (for brevity we write A= 1). Implicit in the previous remark is the fact that a representation 8 with A=l is an interior boundary. We shall state that as soon as we develop a few preliminaries. Consider the vertices of 8 as a partially ordered set under the relation 8y> 8fcif j=>k. A chain will be a maximal totally ordered set of vertices. A normal 8 is called properly nested if for any two vertices 8; and 8k either j =>k, k^j, or j\k. Lemma 1. Suppose 8 is a normal representation with a positive outer point. If X(k) = 1 for k= 1, 2,..., n, then 8 has nonnegative winding number at every point and 8 is properly nested. Proof. It follows from [3, Theorem 4, p. 1090] that 8 has nonnegative winding number at every point. Suppose by way of contradiction that 8 is not properly nested. Then there exist d, and dk such that dk License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1968] EXTENSIONS OF INTERIOR BOUNDARIES 83 If cT*is not dk_x, then d* Theorem. If 8 is a representation with X=l, then 8 is an interior boundary. Proof. If 8mis the smallest vertex in a chain, then it follows from Lemma 2 that 8 has a cut of type I at 8mand that 8* is a representation with A= 1. (See Paragraph 7 of [4].) By Theorems 4 and 5 of [4] we have that S is an interior boundary if and only if 8* is. But 8* has one less vertex than 8 [4, Theorem 6, p. 55]. An induction argument completes the proof. Lemma 3. Let 8 be a normal representation defined on the boundary of a Jordan domain D and suppose fis a properly interior extension of 8 to D. Let 8mbe a terminal vertex of 8. IfTis the interval t'mStS t"m,then there exists a Jordan curve J^D such that J n(D-D) = T, f(J) =f(T), and for some k > 1,f\J is topologicallyequivalent to w = zk on |z| = L Proof. Pick s in T so that t'm License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 84 M. L. MARX [April points of multiplicities p(l), p(2),..., p(k). Note that for any p e Y(8), 2J=i p(j) is equal to one plus the tangential winding number of 8 [2, Theorem 20.1, p. 71]. It follows from this that 8 has an extension which is a local homeomorphism if and only if the tangential winding number of 8 is one; in this case, T(8) is empty. We make the convention that O denotes the empty sequence. When 8 has a local homeomorphism extension, we consider O the only element of Y(8). If p and v are any sequences as above, define p u v to be the sequence p(X),..., p(k), v(l),..., v(l). Define pu <&=p. The following theorem gives an algorithm for constructing p in T(S) with \p\ minimal, where Sisa representation with A= 1. Before stating the theorem we make one definition. Definition. Let 8 be a representation with A=l where 8 is defined on [0, 2-n]. Suppose 8m is the terminal vertex of a chain of 8 of maximal length and assume this length is at least two. Define /={S( | /?=>/} where 8P is the smallest vertex in the chain such that p=>m. Each 8¡ in / is a terminal vertex by the maximality of the length of the chain. Of course, 8mis in 7. For each 8¡ in 7 let 7( = [t[, t'/]. Define 8* to be the representation given by S|([0, 27r]— ij {7¡ | 8, in /}); thus, 8* is obtained by removing from S the loops corresponding to the vertices 8, in /. Suppose we renumber the vertices of 8 so that 8X,..., 8r are the terminal vertices of S not in / and 8r+x,..., 8S the terminal vertices of S in /. We number the vertices of 8* so that 8f is the terminal vertex of 8* corresponding to 8} (new numbering), 1^/gr, and 8f+1 corresponds to the vertex SP of S in the old numbering. (See Figures 1 and 2.) Suppose that 8 has every chain of length 1 and has s such chains. By Lemma 3, 8 has at least í branch points. But the tangential winding number of S is s+1 and so every branch point has multiplicity one. We can thus ignore this case. Theorem 1. Suppose 8 is a representation with X=l with vertices numbered as in the definition above. (1) If p is in Y(8), then \p\^s, where s is the number of terminal vertices of 8. If \p] =s, then there is a 1-1 correspondence between the terminal vertices and the K/)- (2) Suppose p is a sequence of positive integers with \p\ =s. Then p is in Y(8) with p(j) belonging to the terminal vertex 8;, 1 ^j^s, if and only ifp* is in Y(8*) where P*U) = l4J) for lâj^r, P*(r+l)= 2 0*0")-1) i = r + l with p*(J) belongingto 8f, 1 Sj^r+1. Proof. We first dispose of (1) by appealing to Lemma 3. To each terminal vertex there corresponds a Jordan curve7on which/, an extension of 8, is topologi- cally equivalent to w = zk, k> 1, on |z| = 1. There must be a branch point in Ins 7; License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1968] EXTENSIONS OF INTERIOR BOUNDARIES 85 S2 = 8* Figure 2 if \p\ is to be minimal there can be only one, since/can be redefined to be topologi- cally equivalent to w=zk on \z\ < 1 in Ins 7. Now we turn our attention to (2). First suppose p is in F(8) with \p\ minimal. Let/be an extension of 8 to D with branch point structure given by p, where p(j) belongs to the terminal vertex 8¿, 1 SjSs. Let 8, be a terminal vertex in 7 Then there exists a Jordan curve 7 as in Lemma 3 such that, on 7, /is topologically equivalent to w = zk on |z| = l. Note that k = p(j) —1 since p(j) belongs to S;. We may assume that, on 7 u Ins 7, / is topologi- cally equivalent to w = zk on \z\ S 1. Thus, there are arcs X and Y in Ins 7 such that X has end point t'¡, Y has end point t], X and Y intersect only at their other end point, andf(X)=f(Y). Let Dx be the disk bounded by D-D-[t), t'¡] and Xu Y. There exists a map « from Dx onto \z\ S 1 such that « is a homeomorphism on Dx-X- Y, on X, and on Y. Also h(X)=h(Y)={z \ z real, OSzS 1} and, for x in X and y in Y, h(x) = h(y) if and only if f(x) =f(y). Clearly g=fh~1 is well defined, continuous, light everywhere, and open, except possibly on h(X); therefore, g is open on h(X) [5, Theorem 9, p. 336]. The map g is a properly interior extension of License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 86 M. L. MARX [April 8 with the loop corresponding to S; removed. Note that g has the same branch point structure as / except that the branch point of multiplicity p(j) has been replaced by one of multiplicity p(j) —l. We continue this process, removing the loops associated with the vertices in I and obtaining an extension/* of 8* with multiplicities p(l), p(2),..., p(r), p(r+1)— 1,..., p(s) —l. Applying Lemma 3 to 8*+1 yields a Jordan curve7 such that/* |7 is topologically w=zk on \z\ = 1. If z, is the branch point off of multiplicity p(j) belonging to 8; in I, then by the construction above we can assume that there is a z'j which is a branch point of/* of multiplicity p(j)— 1. Since p=>j and p(j) belongs to Sy, we can also assume that z'¡ is in Ins 7. Thus, k = Jisj=r+ x (p(j)~ 1) = P*(r+ 1). We can define/* to be topologically z«* License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1968] EXTENSIONS OF INTERIOR BOUNDARIES 87 which has branch point structure as in (2). Hence, we have proved the theorem if we establish the claim. The claim will be proved by induction on u. If u= 1, the claim is clear. Suppose it true for u— 1 > 1. Without loss of generality take 7 and/(7) to be the unit circle. By the induction hypothesis there is a map g, and arcs y1,..., yu~1 which satisfy the claim with respect to z°a) + ■••+"("-d + 1 and p(l),..., p(u— 1). Let e be an arc in Ins g(J) with one end point at g(tv), which does not intersect any g(y'). By Theorem 1 [1, p. 49], there is an arc e in Ins 7 with one end point at tv, mapping homeo- morphically onto e. Note we are done if p(u) = 0. Assume again without loss of generality that tv=l,e={z\ zreal, O^z^ 1}, andg\e = zpiu)+1. Define/on 7 u Ins 7 by letting/be g(zpW + 1) on {z | 0^|z|^l, 0^argz^27r/(,5(M)-|-l)} and zi,(u)+ 1 on the rest of the unit disk. Then / has the required properties, where yu is e. This establishes the claim and hence, the theorem. Using Theorem 1 we can test whether p is in Y(8) or not, with \p\ minimal, by examining p*. There is an improvement in our situation since 8* has all chains no longer than those of S and at least one strictly shorter. Thus, repeated application S3 = (82)* Figure 3 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 88 M. L. MARX [April of the theorem reduces the problem to examining a representation with every chain of length one. In this case the only admissible p is one which has a branch point of multiplicity one belonging to each terminal vertex. Also, starting with such a representation we can build up to a p in Y(8) by using the theorem in the other direction. Note that at any point there may be several chains which are of maximal length and so there are several choices for which 8* to use. Since the theorem is "if and only if" it is immaterial which S* is chosen. Example. In Figures 1-4 we reduce a representation by the method of the theorem until we obtain the representation S4 (Figure 4). It is clear that this representation has one branch point of multiplicity 2. Working backwards we get r(S3): (3, 1); (2, 2); (1, 3). (The first entry belongs to 8?; the second, to S|.) T(8*): (3, 2); (2, 3); (1, 4). (The first entry belongs to Sf ; the second, to Sf.) r(S): (4, 1, 2), (3, 2, 2), (2, 3, 2), (1, 4, 2); (3, 1, 3), (2, 2, 3), (1, 3, 3); (2, 1, 4), (1, 2, 4). (The first entry belongs to 84; the second, to S5; the third, to S7.) S4 = (S3)* Figure 4 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1968] EXTENSIONS OF INTERIOR BOUNDARIES 4. The cut process. Let 8 be a normal representation of a closed curve. In this paragraph we define representations which are "more like" representations with A=l. We show that these representations determine whether 8 is an interior boundary and, if so, determine the branch point structure of 8. Assume 8 has nonnegative circulation; otherwise, it cannot be an interior boundary. If S has A(;) = 1 for all vertices 8¡, then Theorem 1 gives the desired information about 8 and T(S). So we assume that this is not the case; we select a positive outer point 8(0) and, with respect to that, number the vertices as in Para- graph 2. We select the least q such that X(q)= —1. Suppose P={8¡ \j Figure 5 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 90 M. L. MARX [April Figure 6 Case B(m). Define 8* and S** by 8* = 8q8m(-ß)+ 8(Q8(0(8) 8** = 8(0)8(0(8) + 8m85(j3)+8(08(277)(S). See Figure 8. Also in this case, if there are p nests, then for each r, 1 S r Sp, with t'm< sr define er and £r by e' = 8m8(ur)(ß) + 8(ur)8(sr)(y') + 8(sr)8(ur)(ß) + 8(ur)8(Q(?>) (Figure 9), V = S(0)S(ir)(S)+ 8(sr)8(ur)(ß)+ S(wr)S(ír)(/)+ 8(sr)8q(ß)+ 8(t'Q)8(27t)(8) (Figure 10). Let k be the least positive integer such that t'm License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1968] EXTENSIONS OF INTERIOR BOUNDARIES 91 Case A(2):(i < t', < tl < t"2 mod 8 modS* Figure 7 8" Figure 8 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 92 M. L. MARX [April mod d Figure 9 mod £' Figure 10 mod Figure 11 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1968] EXTENSIONS OF INTERIOR BOUNDARIES 93 such that x8(ur)(a) + 8(ur)x(—jS)is a positively oriented Jordan curve (i.e., a lies to the right of ß). Then define mod er by mod * = 8mx(j3)+ xS(Wr)(«)+ S(Mr)S(ir)(/)+ S(ir)S(Mr)03)+ S(Mr)S(O(8). (See Figure 9.) Similarly, let y be a point of [ß] which is encountered after 8(ur) and such that there are no vertices of 8 between 8(ur) and y. Let a* represent an arc oriented from S(jr) to y with a* to the right of ß. Define mod £r by mod£' = 8(0)8(sr)(8)+ 8(^)8(^)08) + 8(ur)8(sr)(y>) + 8(sr)yC**)+yUß)+ S(OSO)(8). (See Figure 10.) Smooth mod er and mod ÏJ so that they are regular. We leave it to the reader to prove that mod er is normal, mod er is an interior boundary if and only if er is, and r(£r) = r(mod £r). A similar statement is true for £r. Hoping that the reader's patience is not exhausted, we make one more definition. Let 77be a representation with A= 1, suppose it is parametrized by the usual angle parameter 0 S 0 is 277.The outer boundary of 77is a Jordan curve ; suppose it has a positively oriented parametrization (v aug zk)(6) = v(k8), 0úes^~ = 4>((k+1)0-2», T$ï^e^ ^r« J=i,2,...,k. Note that r¡ aug zk has the same topological properties as a representation r¡* with X= 1 whose chains of vertices are described as follows: If 8(l > SÍ2> • • • > 8ir is a chain of 77,then 8X> 82 > ■■ ■ > 8k> 8tl +k> • • ■> 8ir +k is a chain of 77*.(See Figure 12.) Thus, we can apply the results of §3 to 77aug zk. 77' aug z2 (modified) Figure 12 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 94 M. L. MARX [April Theorem 2. Suppose for some m Theorem 3. If for some m Proof. Observe that if e is an interior boundary with some properly interior extension g, then 8** (as defined in Case B(m)) is an interior boundary which has an extension with the same branch points as g. Then the proof again is similar to the proof of Theorem 7' [4, p. 55]. Theorem 4. If for some m License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1968] EXTENSIONS OF INTERIOR BOUNDARIES 95 g2\T2 is topologically zN. However, one traversing of the loop by g2 is lost due to using 72 instead of 72 and taking part of 72 together with L to get a representation of 77. We know that there is a properly interior extension g3 of Theorem 5. Suppose 8 is an interior boundary where Xi£l and let q be the least integer such that X(q)= —1. Then for some m Proof. Suppose /is a properly interior extension of S to D. Since X(q)= -1, by Theorem 1 [1, p. 49], there is an arc A in D with one end point at t"qsuch that A intersects D-D only at t"qand possibly its other end point b; also, f(A)^[ß]. If b is in D, this same Theorem 1 gives that/(¿>)=|8(0) = 8(0). However, an interior mapping cannot map a point of D into the boundary off(D), but 8(0) is an outer point and hence, on the boundary on f(D). We can conclude that b is in D —D. Since /isa local homeomorphism at each point of D-D, it must be that b = t"m for some m and since/(è) is in [ß], m Theorem 6. Suppose 8** is an interior boundary andq is the least integer such that X(q)= —1. Suppose for some m Proof. Let/** be an extension of 8** to D. Suppose a1 describes an arc which intersects [8] only at its end points 8(s}) and 8(u,), 1 SjSp; let it be oriented from 8(u,) to 8(s,). IfKj is the Jordan curve [8(^)8(^)03) + 8(uj)8(sj)(a% then a' should be located so that [y^^Kj u Ins K¡. Let A be the arc on D-D such thatf**\A describes 8\[t'm,t'q] and let B be the arc such that 8\B describes 8m8,(j3).Suppose x'¡ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 96 M. L. MARX [April and y] are in A and x¡ and y¡ are in B, where f**(x¡)=f**(x'¡) = 8(s¡) and f**(y,) = f**(yi)=«(«;). By an argument similar to that in Lemma 3, there is an arc 77, in D with one end point at x,-, the other, z¡, in A u B, and f**(Hj)<^Kj. Let us first suppose that for each/ 1 íkjúp, the other end point z; is in B. Then since/** is properly interior, z¡=y}. Denote by L¡ the Jordan curve formed by 77, and //,' = [x,, y,]. By Theorem 4.2 [6, p. 96],/**|7, is topologically equivalent to the mapping w = zk for some positive integer k. Hence, there exists a properly interior mapping g¡ on 7y u Ins L¡ such that gj\Hj=f** and gj\H'} traces —y'. Also, g} can be taken to have the same branch point structure on 7; u Ins7; as/** does. If we define h to be/** on D- Ui-i (7, u Ins 7,) and g;- on 7; u Ins 7;, 1 èjèp, then by Theorem 9 [5, p. 336], A is properly interior on D; by definition of e, A extends e. Since /z has the same branch point structure as/**, it must be that every p in T(8**) which has an extension/** with every z} in B is in Y(e). We shall soon prove that if some z¡ is in A, then e1 and £J are interior boundaries. Hence, in the case that e1 and V, 1 újúp, fail to be interior boundaries, every extension/** of 8** is as above. We conclude that in this case, r(S**)<= Y(e). Now we show that r(e)cr(S**). Let g be a properly interior extension of e to D. Let Bj = [ut, Vj] be the arc of D-D mapping onto y1, 1 újúp- Choose C¡, an arc in E2 —D, so that C; intersects D —D only at its end points ut and v¡ and so that traversing B¡ from u¡ to vt and then C} from «; to v¡ is in the negative orientation. Let g, be a homeomorphism such that gj\Bt = g and g, maps Q onto [8(^)8(^)03)]. Clearly g¡ can be extended to be a homeomorphism on F3 = /?, u Q u Ins (B¡ u C;). Call this extended map gf. If we define h on 2?=U?«i £/ u D by A|/3= g and h\Ej=gf, then A is a properly interior extension of 8** with the same branch point structure as g. Hence, Y(e)<=Y(8**) and so r(e) = T(8**). Now suppose for some r, 1 Sr^p, that zr is in ^. Since/** is properly interior, it must be that zr=y'r. A similar argument to that used for 77r produces an arc 77^' in D such that yr is one end point, x'r is the other, 77" n (D —D)={x'r,yr}, and f**(H'r)cKr. Let D2 he the Jordan domain bounded by 77r, 77,, 77r', and [x], y'¡]. Let Dx be the component of D —D2 with the pre-image under/** of 8(0) on its boundary; D3, the other component. (We note in passing that for the given/**, there is no Hhj=£r, with one end point in A and the other in B. This follows from the fact that one end point of 77, would be in Dx, the other in D3. This is impossible since 77, n Hr = 0.) Let P be an arc in D2 which intersects Dx only at its end points x'r and yr. Suppose that g is a function defined on P which represents 8(sr)8(ur)(ß) + 8(ur)8(sr)(yr) + 8(sr)8(ur)(ß). If N is the number of pre-images of S(jr) on H¡, then the function g* on the Jordan curve P u Hr' defined by g*\P=g, g*\H'T'=f**, is topologically equivalent to w=zN + 1. Extend g* to be properly interior on EX=P U 77; u Ins (P u 770. Defining g** on Dxv Ex by g**\Dx=f**, g**\Ex=g*, we obtain a properly interior extension of £r. Again we can argue that there are no branch points of/** on Dx— Dt for í = 1,2, 3. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1968] EXTENSIONS OF INTERIOR BOUNDARIES 97 Let v** = v(l),..., v(L) denote the multiplicities of the branch points of g* and let v*=v(L+l),..., v(L*) be the multiplicities of the branch points of/** in Dx. Note that v = v** u v* is in T(£r) and 2f-i v(i) = N. By a similar argument we can produce a properly interior extension of cr with ¿i in r(er) such that p*=p(k +1),..., p(k*) are the multiplicities of the branch points of/** in 7>3and 2f-i /¿(0 = AT»where Af is ths number of pre-images of S(wr)on TTr.We now only have to note that by definition of ATand N,f**\D2 —D2 represents 77mod zM+N~1where 77is as in Theorem 4. Letting £ be the sequence of multiplicities of branch points of/** in D2, we have that £ is in F(r¡ aug ZM+N~1) and the branch point multiplicities of/** are it* u v* u f. The theorem is thus proved. 5. The finiteness of the algorithm. It is now necessary to show that we are in some sense better off by considering the cuts instead of the original representation 8. We have already observed that the branch point structure of S is reflected by that of its cuts. Now we modify the cuts so that they are normal and prove that they have strictly less vertices than 8. Thus, after a finite number of steps, the cut process will terminate, leaving normal representations which either have A= 1 or are Jordan curves. In either case, the branch point structure is determined; in the former case, by Theorem 1. In Case A(m), define mod 8* and mod 8** as in case II'(j) [4, p. 57]. (See Figure 7.) In Case B(m), when there are no nests of 8 with respect to q, define mod 8* and mod 8** as in case II"(j) [4, p. 58]. Also, in Case B(m), when there are nests, e is modified as in case II"(j)- (See Figure 11.) It is not difficult to show 8* is an interior boundary if and only if mod 8*; also, r(S*)=r(mod S*). A similar statement holds for mod 8** and mod e. By Theorem 10 [4, p. 59] we have that mod 8*, mod 8**, mod e, (in the appropriate cases) have Sn —2 vertices, where n is the number of vertices of 8. If there are no nests of 8 with respect to 8„ then we use mod 8* and mod 8** as the cuts. When there are nests, the modification process of [4] would cause 8** to have more vertices than S ; hence the introduction of the cuts er and £r. We have already defined mod er and mod £r and observed that they properly preserve critical point structure. Finally, we show that they have less vertices than 8. Theorem 7. In Case B(m), mod er and mod tj have Sn —2 vertices, where 8 has n vertices. Proof. The proof will be for er ; that for £r is similar. Except for the loop added at 8(ur), mod er traces a subset of [8] and no arc is retraced. The loop at 8(ur) adds one vertex, but it replaces a nest, which had at least one vertex. These facts imply that mod er has Sn vertices. However, 8m and 8, are not vertices of mod er and so the number of vertices of mod er is S « —2. A given representation may have many different cut sequences, each giving rise to a different subset of T(8). We illustrate this by an example. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 98 M. L. MARX Consider the representation S in Figure 6. The cuts which yield interior bound- aries are shown in Figures 7-12. Notice that any p in Y(8) which arises from a cut other than the e, t, cuts in Case B(l) has \p\ ^4. In Case B(l), e1 and Ç1are interior boundaries (see Figures 8 and 9). If p is in T^1) with \p\ minimal, then |/*| = 1. Thus, p* = References 1. M. L. Marx, Normal curves arising from light open mappings of the annulus, Trans. Amer. Math. Soc. 120 (1965), 45-56. 2. M. Morse, Topological methods in the theory of functions of a complex variable, Annals of Mathematics Studies No. 15, Princeton Univ. Press, Princeton, N. J., 1947. 3. C. J. Titus, A theory of normal curves and some applications, Pacific J. Math. 10 (1960), 1083-1096. 4. •—•-, The combinatorial topology of analytic functions on the boundary of a disc, Acta Math. 106 (1961), 45-64. 5. C. J. Titus and G. S. Young, The extension of interiority, with some applications, Trans. Amer. Math. Soc. 103 (1962), 329-340. 6. G. T. Whyburn, Topological analysis, Princeton Univ. Press, Princeton, N. J., 1958. Vanderbilt University, Nashville, Tennessee License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use if k = k* = l.) Similarly let v be in T(mod £r) with \v\ =L* and suppose v(l),..., v(L) belong to the terminal vertex 8(sr) of tj. Define N=J,f=x v(i) and v* = v(L+l),..., v(L*). (Let v* =
ifL=L* = l.)Let7] = S(ir)S(Mr)(S)+ 8(ur)8(sr)(y0- V i & m F(v aug zM+N~x), then p* U v* u f is in Y(8**). Proof. Suppose gx is a properly interior extension of £r to the Jordan domain Dx ; similarly let g2 be a properly interior extension of er to D2. Let s and s' be the pre-images of 8(ir) under gx ; let u and u' be the pre-images of 8(ur) under g2. Let the arcs Tx in Dx and 72 in D2 be as in Lemma 3. There is a point v* in Dx —Dx such that gx(y*)=y. Suppose s is encountered before s' as Dx —Dx is traversed from y* to y* in the positive orientation. We may assume there are no branch points of gx mapping into Ins [8(sr)y(a*)+y8(sr)(-ß)]. This fact and an application of Theorem 1 [1, p. 49] produce an arc 77 in Dx mapping homeomorphically onto [8(sr)y(ß)] where 77 intersects Dx —Dx only at its end points s' and y*. Let Bx be the open disk bounded by 77, Tx, and [y*, s]. Let x* be the point of 7>2— D2 such that g2(x*) = s. Suppose u and u' are such that x* is not in [u, u']. Let z be the pre-image of 8(sr) first encountered as 72 is traced from uto u'. An argument similar to that of the previous paragraph pro- duces an arc K in D2 mapping homeomorphically onto [x8(sr)(ß)], where K inter- sects D2 —D2 only in its end points x* and z. Let B2 be the open disk bounded by K, [«', x*], and T'2, the closure of the component of T2—{z}containing ü'. Assume without loss of generality that Bx and B2 intersect in one and only one point, s' (on Bx) = z (on B2). Let L be an arc joining s and u' which touches Bx u B2 only in those points. Suppose c4 represents 8(sr)8(uT)(8)on L. Then the represen- tation *defined on Tx u 72 u L by c/>*\Tx=gx,*\T'2=g2, *\L=(/>, is equivalent to a representation of r¡ augzM+N_1. This follows since gx\Tx is topologically zM;and M=l. Similarly for v in Y(ir) with |v| minimal, v* =